We consider the Belkale-Kumar cup product ⊙ t \odot _t on H ∗ ( G / P ) H^*(G/P) for a generalized flag variety G / P G/P with parameter t ∈ C m t \in \mathbb {C}^m , where m = dim ( H 2 ( G / P ) ) m=\dim (H^2(G/P)) . For each t ∈ C m t\in \mathbb {C}^m , we define an associated parabolic subgroup P K ⊃ P P_K \supset P . We show that the ring ( H ∗ ( G / P ) , ⊙ t ) (H^*(G/P), \odot _t) contains a graded subalgebra A A isomorphic to H ∗ ( P K / P ) H^*(P_K/P) with the usual cup product, where P K P_K is a parabolic subgroup associated to the parameter t t . Further, we prove that ( H ∗ ( G / P K ) , ⊙ 0 ) (H^*(G/P_K), \odot _0) is the quotient of the ring ( H ∗ ( G / P ) , ⊙ t ) (H^*(G/P), \odot _t) with respect to the ideal generated by elements of positive degree of A A . We prove the above results by using basic facts about the Hochschild-Serre spectral sequence for relative Lie algebra cohomology, and most of the paper consists of proving these facts using the original approach of Hochschild and Serre.